{
  "id": "2207.00313",
  "version": "v1",
  "published": "2022-07-01T10:01:14.000Z",
  "updated": "2022-07-01T10:01:14.000Z",
  "title": "Some Remarks on the Regularized Hamiltonian for Three Bosons with Contact Interactions",
  "authors": [
    "Daniele Ferretti",
    "Alessandro Teta"
  ],
  "comment": "14 pages. To appear in \"Indam Quantum Meetings 22\" proceedings",
  "categories": [
    "math-ph",
    "math.MP",
    "quant-ph"
  ],
  "abstract": "We discuss some properties of a model Hamiltonian for a system of three bosons interacting via zero-range forces in three dimensions. In order to avoid the well known instability phenomenon, we consider the so-called Minlos-Faddeev regularization of such Hamiltonian, heuristically corresponding to the introduction of a three-body repulsion. We review the main concerning results recently obtained. In particular, starting from a suitable quadratic form $Q$, the self-adjoint and bounded from below Hamiltonian $\\mathcal H$ can be constructed provided that the strength $\\gamma$ of the three-body force is larger than a threshold parameter $\\gamma_c$. Moreover, we give an alternative and much simpler proof of the above result whenever $\\gamma > \\gamma'_c$, with $\\gamma'_c$ strictly larger than $\\gamma_c$. Finally, we show that the threshold value $\\gamma_c$ is optimal, in the sense that the quadratic form $Q$ is unbounded from below if $\\gamma<\\gamma_c$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-01T10:01:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "81Q10",
      "81Q15",
      "70F07",
      "46N50"
    ],
    "keywords": [
      "contact interactions",
      "regularized hamiltonian",
      "instability phenomenon",
      "minlos-faddeev regularization",
      "three-body repulsion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}